Uniform Quasi-Concavity in Probabilistic Constrained ... - Rutcor

R u t c o r
Research
R e p o r t
Uniform Quasi-Concavity in
Probabilistic Constrained
Stochastic Programming
András Prékopa a Kunikazu Yoda b
Munevver Mine Subasi c
RRR 6-2010, April, 2010
RUTCOR
Rutgers Center for
Operations Research
Rutgers University
640 Bartholomew Road
Piscataway, New Jersey
08854-8003
Telephone: 732-445-3804
Telefax: 732-445-5472
Email: rrr@rutcor.rutgers.edu
http://rutcor.rutgers.edu/∼rrr
a RUTCOR, Rutgers Center for Operations Research, 640 Bartholomew
Road, Piscataway, NJ 08854-8003; prekopa@rutcor.rutgers.edu
b RUTCOR, Rutgers Center for Operations Research, 640 Bartholomew
Road, Piscataway, NJ 08854-8003; kyoda@rutcor.rutgers.edu
c RUTCOR, Rutgers Center for Operations Research, 640 Bartholomew
Road, Piscataway, NJ 08854-8003; msub@rutcor.rutgers.edu

Rutcor Research Report
RRR 6-2010, April, 2010
Uniform Quasi-Concavity in Probabilistic
Constrained Stochastic Programming
András Prékopa Kunikazu Yoda Munevver Mine Subasi
Abstract. A probabilistic constrained stochastic programming problem is considered,
where the underlying problem has linear constraints with random technology
matrix. The rows of the matrix are assumed to be stochastically independent and
normally distributed. For the convexity of the problem the quasi-concavity of the
constraining function is needed that is ensured if the factors are uniformly quasiconcave.
In the paper a necessary and sufficient condition is given for that property
to hold. It is also shown, through numerical examples, that such a special problem
still has practical application in optimal portfolio construction.
Acknowledgements:
CMMI-0856663.
This research was supported by National Science Foundation Grant

Page 2 RRR 6-2010
1 Introduction
The stochastic programming problem, termed programming under probabilistic constraint
can be formulated in the following way:
minimize f(x) (1.1)
subject to h 0 (x) = P (g i (x, ξ) ≥ 0, i = 1, . . . , r) ≥ p
h i (x) ≥ 0, i = 1, . . . , m,
where x ∈ R n , ξ ∈ R q , f(x), g i (x, y), i = 1, . . . , r, h i (x), i = 1, . . . , m are some functions and
p is a fixed large probability, e.g., p = 0.8, 0.9, 0.95, 0.99. In many application the stochastic
constraints have the form ξ − T x ≥ 0 and the probabilistic constraint specializes as
h 0 (x) = P(T x ≤ ξ) ≥ p. (1.2)
For the case of continuously distributed random vector ξ general theorems are available
to ensure the convexity of the set determined by the probabilistic constraint in (1.1). For
example, if g i , i = 1, . . . , r are concave or at least quasi-concave in all variables and ξ has
a logconcave p.d.f., then the function h 0 (x) is logconcave and the set {x | h 0 (x) ≥ p} is
convex (see, e.g. [12, 14]). This implies that if ξ has the above-mentioned property, then
the set determined by the constraint (1.2) is convex. Many applications of the model with
probabilistic constraint (1.2) have been carried out, for the cases of some special continuous
multivariate distributions such as normal, gamma, and Dirichlet, and problem solving
packages have been developed [14, 3, 6, 15].
The solution of problems where ξ in (1.2) is a discrete random vector is more recent. The
key concept here is that of a p-efficient point, introduced in [11] and further developed and
used in [2, 4, 8, 13].
For the case of a random T in the constraint (1.2), few results are known. The earliest
papers dealing with random matrix T in probabilistic constraint are [7, 9]. In these papers,
however, there is only one stochastic constraint and to establish the concavity of the set
{x | P(T x ≤ ξ) ≥ p} is relatively easy (see, the proof of Lemma 2.2).
The first paper where convexity theorems are presented for the set of feasible solution
and random matrix T has more than one row, is [10]. If T has more than one row, then even
if they are independent, it is not easy to ensure the convexity of the set {x | P(T x ≤ d) ≥ p}.
The papers [12] and [5] can be mentioned, where progress in this direction has been made.
The problem is that the products or sums of quasi-concave functions are not quasi-concave,
in general. We briefly recall the main results of the paper [10] (see also [12] pp.312–314).
Theorem 1.1. Let ξ be constant and T a random matrix with independent, normally distributed
rows (or columns) such that their covariance matrices are constant multiples of each
other. Then h(x) = P(T x ≤ ξ) is a quasi-concave function on the set {x | h(x) ≥ 1/2}.
We introduce a special class of quasi-concave functions.

RRR 6-2010 Page 3
Definition 1.1 (Uniformly quasi-concave functions). Let h 1 (x), . . . , h r (x) be quasi-concave
functions on a convex set E ∈ R n . We say that they are uniformly quasi-concave functions
if for any x, y ∈ E either
min(h i (x), h i (y)) = h i (x), i = 1, . . . , r
or
min(h i (x), h i (y)) = h i (y), i = 1, . . . , r.
Obviously, the sum of uniformly quasi-concave functions, on the same set, is also quasiconcave
and if the functions are also nonnegative, then the same holds for their product as
well. The latter property is used in the next section, where we prove our main result.
In this paper we look at probabilistic constraints of the type
P(T x ≤ b) ≥ p, (1.3)
where T is a random matrix that has independent, normally distributed rows and b is a
constant vector. The constraining function in (1.3) is the product of special quasi-concave
functions and we show that the uniform quasi-concavity of the factors implies that the
covariance matrices of the rows are constant multiples of each other. Section 2 and 3 are
devoted to this. In section 4 we show that this very special type of probabilistic constraint is
still applicable to solve portfolio optimization problems. We present some numerical results
in this respect.
2 Preliminary Results
First we provide a necessary condition for continuously differentiable and uniformly quasiconcave
functions h 1 (x), . . . , h r (x) on an open convex set.
Lemma 2.1. If h 1 (x), . . . , h r (x) are continuously differentiable and uniformly quasi-concave
on an open convex set E, then any nonzero gradients ∇h i (x), ∇h j (x) are positive multiples
of each other, i.e., for any i, j ∈ {1, . . . , r}, there exists a positive-valued function α ij (x) =
1/α ji (x) > 0 defined on E ij = {x ∈ E | ∇h i (x) ≠ 0, ∇h j (x) ≠ 0} = E ji such that for all
x ∈ E ij we have
∇h i (x) = α ij (x)∇h j (x) (2.1)
Proof. We show that (2.1) holds for all x ∈ E ij by contradiction. Suppose that for some
x ∈ E ij we cannot find an α ij (x) > 0 satisfying (2.1). Without loss of generality we assume
that i = 1, j = 2.
From Farkas Lemma, either one of the following two systems has a solution
(i) ∇h 2 (x) T d ≤ 0, ∇h 1 (x) T d > 0
(ii) ∇h 1 (x) = λ∇h 2 (x), λ ≥ 0

Page 4 RRR 6-2010
First, note that since ∇h 1 (x) ≠ 0 and ∇h 2 (x) ≠ 0, λ = 0 cannot be a solution of (ii).
Also, λ > 0 cannot be a solution of (ii), otherwise we can define α 12 (x) = λ > 0. Hence,
(i) has a solution d 1 . Similarly, since ∇h 2 (x) = α 21 (x)∇h 1 (x) does not hold for any defined
value of α 21 (x) = 1/α 12 (x) > 0 by the assumption, (i) with 1 and 2 interchanged has a
solution d 2 . So we have
Let d := d 1 − d 2 . Then it follows that
∇h 2 (x) T d 1 ≤ 0, ∇h 1 (x) T d 1 > 0,
∇h 1 (x) T d 2 ≤ 0, ∇h 2 (x) T d 2 > 0.
∇h 1 (x) T d > 0, ∇h 2 (x) T d < 0. (2.2)
Note that d ≠ 0. By the use of finite Taylor series expansions we can write:
h 1 (x + εd) = h 1 (x) + (∇h 1 (x) T d)ε + o(ε), (2.3)
h 2 (x + εd) = h 2 (x) + (∇h 2 (x) T d)ε + o(ε). (2.4)
Since E 12 is an open set, we can select ε > 0 small enough so that
∃y := x + εd ∈ E 12 , y ≠ x, h 1 (y) > h 1 (x), h 2 (y) < h 2 (x)
Hence h 1 (x), . . . , h r (x) are not uniformly quasi-concave, which is a contradiction.
For r = 1, let us consider the function
h(x) = P(T x ≤ b), (2.5)
where T is a random row vector and b is a constant. The following lemma was first proved
by Kataoka [7] and van de Panne and Popp [9]. See also Prékopa [12].
Lemma 2.2 ([7, 9]). If T has normal distribution, then the function h(x) is quasi-concave
on the set
{
x | P(T x ≤ b) ≥ 1 }
.
2
Proof (from [14], pp 284-285). We prove the equivalent statement: for any p ≥ 1/2 the set
{x | P(T x ≤ b) ≥ p} (2.6)
is convex.
Let Φ(t) denote the c.d.f. of the one-dimensional standard normal distribution and let
µ = E(T T ) and C = E((T T − µ)(T T − µ) T ) denote the mean vector and the covariance

RRR 6-2010 Page 5
matrix of T , respectively. For any x such that x T Cx > 0,
is equivalent to
h(x) = P(T x ≤ b)
( (T − µ T )x
= P √
xT Cx
≤ b − )
µT x
√
xT Cx
( ) b − µ T x
= Φ √ ≥ p
xT Cx
µ T x + Φ −1 (p) √ x T Cx ≤ b. (2.7)
Since Φ −1 (p) ≥ 0 and √ x T Cx is a convex function of x on R n , it follows that inequality (2.7)
determines a convex set.
For any x such that x T Cx = 0, T x = µ T x with probability 1. Since p > 0 it follows that
is equivalent to
The set of x determined by (2.8) is convex.
h(x) = P(T x ≤ b) = P(µ T x ≤ b) ≥ p
µ T x ≤ b. (2.8)
Let r be an arbitrary positive integer and introduce the function:
h i (x) = P(T i x ≤ b i ), i = 1, . . . , r, (2.9)
where each row vector T i , i = 1, . . . , r has normal distribution with mean vector µ i = E(T T
and covariance matrix C i = E((Ti
T − µ i )(Ti
T − µ i ) T ), and b = (b 1 , . . . , b r ) T is constant.
Suppose b i > 0, i = 1, . . . , r. Let us define set E as follows:
E is convex.
E ⊃ B ⊃ {0} for some open set B. (2.10)
Each h i (x), i = 1, . . . , r is quasi-concave on E.
i )
One example of such E is
E =
r⋂
i=1
{
x | h i (x) ≥ 1 }
. (2.11)
2
Note that by lemma 2.2, h i (x) is quasi-concave on the convex set E i = {x | h i (x) ≥ 1/2} and
that for sufficiently small open ball B ɛ (0) = {x | ‖x‖ < ɛ} of the origin, h i (x) ≥ 1/2, ∀x ∈
B ɛ (0), thus E i ⊃ B ɛ (0). Also note that the intersection of convex sets is a convex set. If

Page 6 RRR 6-2010
rows T 1 , . . . , T r of T are independent and h 1 (x), . . . , h r (x) are uniformly quasi-concave, then
h(x) = P(T i x ≤ b i , i = 1, . . . , r) = ∏ r
i=1 P(T i ≤ b i ) = h 1 (x) · · · h r (x) is quasi-concave on E.
Suppose b i > 0 and C i is positive definite for all i ∈ {1, . . . , r}.
Since
⎧ ( )
⎨ bi − µ T i x
Φ √ for x ≠ 0,
h i (x) = xT C
⎩
i x
P(0 ≤ b i ) = 1 for x = 0.
lim h i(x) = lim Φ(t) = 1 = h i (0),
x→0 t→∞
h i (x) is continuous at x = 0. Let us calculate the gradient of h i (x) for x ∈ int (E) \ {0}.
( )
bi − µ T i x
∇h i (x) = ∇Φ √
xT C i x
( )
bi − µ T i x
= φ √ ∇ b i − µ T i x
√
xT C i x xT C i x
( ) √
bi − µ T i x − xT C
= φ √ i xµ i − (b i − µ T i x)C i x/ √ x T C i x
xT C i x
x T C i x
= −φ
(
bi − µ T i x
√
xT C i x
(2.12)
) (x T C i x)µ i + (b i − µ T i x)C i x
(x T C i x) 3/2 , (2.13)
where φ(t) is the p.d.f. of the one-dimensional standard normal distribution.
φ(t) = √ 1 exp
(− 1 )
2π 2 t2 .
For any fixed x ≠ 0, we have
(
lim ∇h i (εx) = − lim φ
ε↓0 ε↓0
= 0.
b i
ε √ x T C i x −
) { µT i x (x T C
√ i x)µ i − (µ T i x)C i x
xT C i x ε √ +
x T C i x
}
b i C i x
ε 2 (x T C i x) 3/2
Hence lim x→0 ∇h i (x) = 0 and ∇h i (x) is continuous at x = 0. Therefore h 1 (x), . . . , h r (x) are
continuously differentiable on the open convex set int (E).
3 The Main Result
In what follows we make use of the following theorem [1] from linear algebra:
Theorem 3.1 (Simultaneous Diagonalization of Two Matrices). Given two real symmetric
matrices, A and B, with A positive-valued definite, there exists a nonsingular matrix U such

RRR 6-2010 Page 7
that
U T AU = I, U T BU = diag(λ 1 , λ 2 , . . . , λ n ) = ⎜
⎝
O
⎛
λ 1
λ 2
. ..
⎞
O
⎟
⎠
λ n
(3.1)
In the next theorem we present our main result.
Theorem 3.2. Suppose b i > 0 and C i is positive definite for i ∈ {1, . . . , r}. The functions
h 1 (x), . . . , h r (x) defined by (2.9) (in this case, (2.12)) are uniformly quasi-concave on a
convex set E satisfying (2.10) if and only if each C i is a constant multiple of a covariance
matrix C, and
µ 1
b 1
= · · · = µ r
b r
.
Proof. Sufficiency (⇐) is obvious, so we only show necessity (⇒). It is enough to show
that C 1 , C 2 are constant multiples of each other and that µ 1 /b 1 = µ 2 /b 2 for r ≥ 2. h i (x) is
continuously differentiable on the open convex set int (E). From (2.13) we have for x ≠ 0
( )
x T bi − µ T i x b
∇h i (x) = −φ √ √ i
xT C i x xT C i x < 0.
Thus ∇h i (x) ≠ 0 for x ≠ 0. We know lim x→0 ∇h i (x) = 0. Let E ′ := int (E) \ {0}. Then
E ′ = {x ∈ int (E) | ∇h i (x) ≠ 0, i ∈ {1, . . . , r}}. From Lemma 2.1 and (2.13), there is a
positive function α 12 (x) > 0 such that for all x ∈ E ′ we have
(x T C 1 x)µ 1 + (b 1 − µ T 1 x)C 1 x = α 12 (x) { (x T C 2 x)µ 2 + (b 2 − µ T 2 x)C 2 x } (3.2)
For small ε > 0 and x ∈ E ′ , let us replace x with εx ∈ E ′ in (3.2) and divide by ε for both
sides of the equation.
ε(x T C 1 x)µ 1 + (b 1 − εµ T 1 x)C 1 x = α 12 (εx) { ε(x T C 2 x)µ 2 + (b 2 − εµ T 2 x)C 2 x } (3.3)
Taking the limit of the both sides of (3.3) as ε → 0 we obtain
Since 0 < x T C 1 x < ∞, 0 < x T C 2 x < ∞ for x ∈ E ′ , the limit
exists and 0 < α ′ 12(x) < ∞. Thus we have
b 1 C 1 x = (lim
ε→0
α 12 (εx))b 2 C 2 x. (3.4)
lim α 12(εx) = b 1x T C 1 x
ε→0 b 2 x T C 2 x =: α′ 12(x) (3.5)
b 1 C 1 x = α ′ 12(x)b 2 C 2 x for x ∈ E ′ . (3.6)

Page 8 RRR 6-2010
Since C 1 and C 2 are symmetric and C 2 is positive definite, from Theorem 3.1 there is a
nonsingular matrix U such that
U T C 1 U = D, U T C 2 U = I,
where D = diag(λ 1 , . . . , λ r ) is a diagonal matrix. Let y := U −1 x and F := {U −1 x | x ∈ E ′ }.
Since U is nonsingular, F is a neighborhood of the origin 0, and 0 /∈ F .
For all y ∈ F we have by multiplying U T from left to (3.6)
which implies that
b 1 Dy = α 12(Uy)b ′ 2 y
⎡ ⎤
⎡ ⎤
λ 1 y 1
y 1
⎢ ⎥
⇒ b 1 ⎣ . ⎦ = α 12(Uy)b ′ ⎢ ⎥
2 ⎣ . ⎦
λ r y r y r
0 < α ′ 12(x) = α ′ 12(Uy) = b 1λ 1
b 2
is constant. Therefore we have from (3.6)
= · · · = b 1λ r
b 2
=: α ′ 12
Let us plug (3.7) into (3.2).
C 1 = α 12
′ b 2
C 2 . (3.7)
b 1
x T C 2 x (α 12b ′ 2 µ 1 − α 12 (x)b 1 µ 2 )
{
}
+ (α 12 ′ − α 12 (x))b 1 b 2 − (α 12b ′ 2 µ 1 − α 12 (x)b 1 µ 2 ) T x C 2 x = 0. (3.8)
Multiplying (3.8) by x T from left we obtain
If we substitute (3.9) into (3.8), we get
{α ′ 12 − α 12 (x)} b 1 b 2 x T C 2 x = 0 ⇒ α 12 (x) = α ′ 12. (3.9)
x T C 2 x(b 2 µ 1 − b 1 µ 2 ) = x T (b 2 µ 1 − b 1 µ 2 )C 2 x. (3.10)
Let us introduce w := U T (b 2 µ 1 − b 1 µ 2 ). Since x = Uy we have
(y T y)w = (y T w)y
⎡ ⎤
⎡ ⎤
w 1
⎢ ⎥
⇒ ⎣ . ⎦ = y y 1
1w 1 + · · · + y r w r ⎢ ⎥
y1 2 + · · · + yr
2 ⎣ . ⎦ (3.11)
w r y r
Since (3.11) holds for y = ε[0, 1, . . . , 1] T , . . . , y = ε[1, . . . , 1, 0] T ∈ F for some small ε > 0, it
follows that
w 1 = 0, . . . , w r = 0 ⇒ w = 0 ⇒ µ 1
b 1
= µ 2
b 2
. (3.12)

RRR 6-2010 Page 9
4 Application in Portfolio Optimization
In this section we look at a probabilistic constrained stochastic programming problem, where
the probabilistic constraint is of type (1.2). We assume that T has independent, normally distributed
rows and the factors in the product ∏ K
k=1 P(T kx ≤ b k ) are uniformly quasi concave.
The problem is special, but still can be applied, e.g., in portfolio optimization.
Consider n assets and K consecutive periods. Let us introduce the following notations:
for k = 1, ..., K
T k : random loss of the assets during the k-th period
µ k = E[T T k ] : expected loss
C k = E[(T T k − µ k )(T T k − µ k ) T ] : covariance matrix of T k .
We assume that T k , k = 1, .., K, are independent and normally distributed random vectors
and µ k ≤ 0, k = 1, ..., K. We also assume that the time window of the K periods is relatively
short and a linear trend for the expectations prevails. Formally, our assumptions are:
µ 1 = µ and µ k+1 = αµ k , k = 1, . . . , K − 1 (4.1)
C 1 = C and C k+1 = α 2 C k , k = 1, . . . , K − 1. (4.2)
For the first period, we consider the portfolio optimization problem formulated by Kataoka
[7]:
(Problem 1):
minimize b
subject to Φ
( ) b − µ T x
√ ≥ p
xT Cx
n∑
x j = 1
j=1
x j ≥ 0 for j = 1, . . . , n.
For the k-th period (k ∈ {2, . . . , K}), we consider the following problem.
(Problem k): minimize b 1
k∏
subject to Φ
i=1
( )
bi − µ T i x
√ ≥ p
xT C i x
n∑
x j = 1
j=1
b i+1 = α b i for i = 1, . . . , k − 1
x j ≥ 0
b 1 ≥ 0 .
for j = 1, . . . , n

Page 10 RRR 6-2010
A related model is presented in [17], where individual probabilistic constraints are taken
for more than one part of the distribution.
By Theorem 3.2 the functions h 1 (x), . . . , h K (x) defined by (2.12) are uniformly quasiconcave
on the convex set E := ∩ K k=1 {x | h k(x) ≥ 1/2} = ∩ K k=1 {x | b k ≥ µ T k x} = {x | b K ≥
µ T K x}, and hence h(x) = ∏ K
k=1 h k(x) is quasi-concave on E. Since set {x | h(x) ≥ p, x ∈ E}
is convex, the set of feasible solutions of (Problem k) is convex.
Below we present a numerical example for the application of the above model. We take
the initial expectations and covariance matrix from past history data but then proceed to
obtain those values in accordance with the assumption formulated in the model.
Numerical Example.
Assets “Dow, S&P500, Nasdaq, NYSECI, 10YrBond” are obtained from Yahoo! Finance
(http://finance.yahoo.com) and assets “Oil, Gold, Silver, EUR/USD” are obtained from
Dukascopy (http://www.dukascopy.com).
We consider the expected values and the covariance matrix of the daily losses of the nine
assets in May 2009. The data is shown in Table 1 and Table 2.
Gold Silver Nasdaq S&P500 Oil EUR/USD 10YrBond Dow NYSECI
-1.253 -3.008 -0.149 -0.711 -1.379 -0.82 -1.052 -0.546 -1.069
Table 1: Expected losses in May 2009
Gold Silver Nasdaq S&P500 Oil EUR/USD 10YrBond Dow NYSECI
Gold 5.159 7.228 -1.437 1.492 3.989 2.764 -5.25 1.198 2.231
Silver 7.228 19.441 -0.785 6.454 10.143 4.94 -5.198 5.343 9.061
Nasdaq -1.437 -0.785 15.084 11.202 1.562 0.974 -0.767 9.754 12.424
S&P500 1.492 6.454 11.202 16.238 10.709 4.223 -4.735 14.794 20.058
Oil 3.989 10.143 1.562 10.709 21.249 4.087 -5.719 10.043 15.451
EUR/USD 2.764 4.94 0.974 4.223 4.087 4.375 -3.255 3.764 5.996
10YrBond -5.25 -5.198 -0.767 -4.735 -5.719 -3.255 38.003 -4.564 -4.928
Dow 1.198 5.343 9.754 14.794 10.043 3.764 -4.564 13.981 18.446
NYSECI 2.231 9.061 12.424 20.058 15.451 5.996 -4.928 18.446 25.706
Table 2: Covariance Matrix in May 2009
We assume that in the consecutive periods the expected returns are increased by α = 1.01
(1%) and the covariance matrix is increased by α 2 = (1.01) 2 . The values of the nine assets
obtained by the use of (Problem k), k = 1, ..., 5 are given in Table 3.

RRR 6-2010 Page 11
Gold Silver Nasdaq S&P500 Oil EUR/USD 10YrBond Dow NYSECI
(Problem 1) 0.5246 0.0422 0.1267 0 0.0102 0.1433 0.1531 0 0
(Problem 2) 0.5350 0 0.1342 0 0.0103 0.1718 0.1487 0 0
(Problem 3) 0.5266 0 0.1379 0 0.0076 0.1804 0.1475 0 0
(Problem 4) 0.5218 0 0.1399 0 0.0063 0.1849 0.1471 0 0
(Problem 5) 0.5188 0 0.1414 0 0.0053 0.1878 0.1467 0 0
References
Table 3: Values of nine assets, May 2009
[1] R. E. Bellman, Introduction to matrix analysis, Society for Industrial Mathematics
(1987) 58–59.
[2] P. Beraldi and A. Ruszczyński, A Branch and Bound Method for Stochastic Integer
Problems under Probabilistic Constraints, Optimization Methods and Software, 17, 3
(2002) 359–382.
[3] I. Deák, Solving stochastic programming problems by successive regression approximations
– numerical results, Dynamic stochastic optimization, Lecture notes in economics
and mathematical systems, Springer 2003, 209–224.
[4] D. Dentcheva and A. Prékopa and A. Ruszczyński, Concavity and efficient points of
discrete distributions in probabilistic programming, Mathematical Programming, 89, 1
(2000) 55–77.
[5] H. Henrion and C. Strugarek, Convexity of chance constraints with independent random
variables, Computational Optimization and Applications, 41, 2, (2008) 263–276.
[6] P. Kall and J. Mayer, Stochastic Linear Programming: Models, Theory, and Computation,
Springer 2005.
[7] S. Kataoka, A Stochastic Programming Model, Econometrica 31 (1963), 181–196.
[8] J. Luedtke and S. Ahmed and G. L. Nemhauser, An integer programming approach
for linear programs with probabilistic constraints, Mathematical Programming, 122, 2
(2010) 247–272.
[9] C. van de Panne, W. Popp, Minimum Cost Cattle Feed under probabilistic protein
constraints, Management Sciences, 9, 3 (1963) 405–430.
[10] A. Prékopa, Programming under Probabilistic Constraints with a Random Technology
Matrix, Mathematische Operationsforschung und Statistik, Ser. Optimization 5 (1974),
109–116.

Page 12 RRR 6-2010
[11] A. Prékopa, Dual method for the solution of a one-stage stochastic programming problem
with random RHS obeying a discrete probability distribution, ZOR - Methods and
Models of Operations Research, 34 (1990) 441–461.
[12] A. Prékopa, Stochastic Programming, Kluwer Academic Publishers, Dordtecht, Boston,
1995.
[13] A. Prékopa and B. Vizvári and T. Badics, Programming under probabilistic constraint
wit discrete random variable, New trends in mathematical programming, F. Giannessi
and T. Rapcsák and S. Komlósi, Kluwer Academic Publishers, (1998) 235–255.
[14] A. Prékopa, Probabilistic Programming, Stochastic Programming, Handbooks in Operations
Research and Management Science, A. Ruszczyński and A. Shapiro, Elsevier,
10 (2003) 207–351.
[15] T. Szántai, A computer code for solution of probabilistic constrained stochastic programming
problems, Numerical Techniques for Stochastic Optimization, Y. Ermoliev
and R. J.-B. Wets, Springer, New York (1988) 229–235.
[16] S. Uryasev, Probabilistic Constrained Optimization: Methodology and Applications,
Kluwer Academic Publishers, Boston, 2000.
[17] K. Yoda and A. Prékopa, Optimal portfolio selection based on multiple Value at Risk
constraints, to appear, RUTCOR research report, Rutgers Center for Operations Research,
2010.